上海交通大学：《数学的天空》课程教学资源（讲义）第六讲 七个百万美元千禧年问题简介

Figure1:克莱数学研究所 课程网站 http://math.sjtu.edu.cn/course/skymath/ 第六讲七个百万美元千禧年问题简介 2000年5月24日，美国克莱数学研究所(Clay Mathematics Insti-- tute)的科学顾问委员会把庞加莱猜想列为七个“千禧难题”(Millennium Prize Problems)之一，这七道问题被研究所认为是“重要的经典问题， 经许多年仍未解决。”克雷数学研究所的董事会决定建立七百万美元的 大奖基金，每个“千年大奖问题”的解决都可获得百万美元的奖励（因 此Millennium Prize Problems又被称为Million Prize Problems)。另外六个 “千年大奖问题”分别是： 1.P与NP问题(P versus NP problem) 2.霍奇猜想(Hodge conjecture) 3.庞加莱猜想(Poincare conjecture) 4.黎曼假设(Riemann hypothesis) 5.杨一米尔斯理论(Yang-Mills existence and mass gap) 6.纳维-斯托克斯方程(Navier-Stokes existence and smoothness,简 称NS方程) 7.BSD(Birch and Swinnerton-Dyer conjecture) 1

Figure 1: ➂✹ê➷ï➘↕ ➅➜✤Õ http://math.sjtu.edu.cn/course/skymath/ ✶✽ù Ô❻③✙④✄❩✩❝➥❑④✵ ✥✥2000❝5✛24❋➜④■➂✹ê➷ï➘↕(Clay Mathematics Insti￾tute)✛❽➷✚➥➈✡➡r✡❭✹ß➂✎➃Ô❻✴❩✩❏❑✵ (Millennium Prize Problems)❷➌➜ ùÔ✗➥❑✚ï➘↕❅➃➫✴➢❻✛➨❀➥❑➜ ➨◆õ❝❊➍✮û✧✵➂❳ê➷ï➘↕✛➶➥➡û➼ïáÔ③✙④✄✛ ➀ø➘✼➜ ③❻✴❩❝➀ø➥❑✵✛✮ûÑ➀➻✚③✙④✄✛ø②(Ï ❞ Millennium Prize Problemsq✚→➃ Million Prize Problems)✧✱✠✽❻ ✴❩❝➀ø➥❑✵➞❖➫➭ 1. P❺NP➥❑(P versus NP problem) 2. ➾Ûß➂(Hodge conjecture) 3. ✡❭✹ß➂(Poincar´e conjecture) 4. ✐ù❜✗(Riemann hypothesis) 5. ✌➝➆✏❞♥Ø(YangõMills existence and mass gap) 6. ❇➅-❞÷➂❞➄➜(NavierõStokes existence and smoothness➜④ →NS➄➜) 7. BSDß➂(Birch and Swinnerton-Dyer conjecture) 1

Set: 1821243357-14-15-32 -56 Some Subsets: 18=18 18-14=4 57-15-32=-10 21+24+57-14-32-56=0 Figure 2:P=NP? P与NP问题 P={Questions for which some algorithm can provide an answer in poly- nomial time} NP=fQuestions for which some an answer can be verified in polynomial time} Question(1971):P=NP? Stephen Arthur Cook,1939.12.14-,美国-加拿大计算机学家、数学家. 2

Figure 2: P=NP? P❺NP➥❑ P={Questions for which some algorithm can provide an answer in poly￾nomial time} NP={Questions for which some an answer can be verified in polynomial time} Question(1971): P=NP? Stephen Arthur Cook➜1939.12.14-➜④■-❭❁➀❖➂➴➷❬✦ê➷❬. 2

Figure3:Hodge猜克 Hodge猜克 在非奇异复射影代数簇上，任一霍奇类是代数闭链类的有理线性组合 (Hodge Conjecture.Let X be a projective complex manifold.Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.) William Vallance Douglas Hodge FRS,1903.6.17-1975.7.7,苏格兰 几何学家 3

Figure 3: Hodgeß➂ Hodge ß➂ ✥✸➎Û➱❊✓❑➇êqþ, ❄➌➾Û❛➫➇ê✹ó❛✛❦♥❶✺⑤Ü. (Hodge Conjecture. Let X be a projective complex manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.) William Vallance Douglas Hodge FRS➜1903.6.17õ1975.7.7➜⑨❶❂ ❆Û➷❬. 3

月alf Hofmann The Thermodynamicsof Quantm Yang-Mills Theory 的时Ap南a 的wer Figure4:杨一米尔斯理论 杨一米尔斯理论 杨一米尔斯存在性与质量间隙.对任意紧单规范群G,证明R4上存在质 量间隙为正的非平凡量子Yang~Mills理论 (Yang-Mills Existence and Mass Gap.Prove that for any com- pact simple gauge group G,a non-trivial quantum Yang-Mills theory exists on R4 and has a mass gap 6>0.) Robert L.Mills,1927.4.15-1999.10.27,美国物理学家. 4

Figure 4: ✌➝➆✏❞♥Ø ✌➝➆✏❞♥Ø ✌➝➆✏❞⑧✸✺❺➓þ♠❨. é❄➾❀ü✺❽✰G➜②➨R 4 þ⑧✸➓ þ♠❨➃✔✛➎➨❹þ❢YangõMills♥Ø. (YangõMills Existence and Mass Gap. Prove that for any com￾pact simple gauge group G, a non-trivial quantum YangõMills theory exists on R 4 and has a mass gap δ > 0.) Robert L. Mills➜1927.4.15õ1999.10.27➜④■Ô♥➷❬. 4

Figure5:纳维-斯托克斯方程 纳维-斯托克斯方程 PDi=-V(p)+V.T+f DU 纳维-斯托克斯方程是否存在光滑的解？ 纳维-斯托克斯方程始于19世纪，但人类至今对其所知甚少.无论是微 风还是湍流，都可以通过理解Navier-Stokes方程的解，来对，们进行解释 和预言. Claude-Louis Navier,1785.2.10-1836.8.21,美国工程师、物理学家. Sir George Gabriel Stokes,1st Baronet FRS,1819.8.13-1903.2.1, 剑桥大学教授，著名数学家、物理学家 5

Figure 5: ❇➅-❞÷➂❞➄➜ ❇➅-❞÷➂❞➄➜ ρ Dv Dt = −∇(p) + ∇ · T + f ❇➅-❞÷➂❞➄➜➫➘⑧✸✶✇✛✮? ❇➅-❞÷➂❞➄➜➞✉19➢❱➜✂❁❛➊✽éÙ↕⑧✩✟. ➹Ø➫❻ ➸❸➫ë✻➜Ñ➀➧Ï▲♥✮Navier-Stokes➄➜✛✮➜✺é➜❶❄✶✮➸ Úýó. Claude-Louis Navier➜1785.2.10õ1836.8.21➜④■ó➜➇✦Ô♥➷❬. Sir George Gabriel Stokes➜1st Baronet FRS➜1819.8.13õ1903.2.1➜ ê①➀➷✓➬➜❮➯ê➷❬✦Ô♥➷❬. 5

p=2.2 3p=+1,0 -I0.0L. 1 e=l-1-IT Figure6:BSD猜克 Birch and Swinnerton-Dyer conjecture 设L(C,s)=Πp2s(1-apps+p-2)1. Birch and Swinnerton-Dyer Conjecture.The Taylor expansion of L(C,s) at s=1 has the form L(C,s)=c(s-1)"+higher order terms with c0 and r rank(C(Q)). Bryan John Birch F.R.S.,1931.9.25,英国数学家 Sir Henry Peter Francis Swinnerton-Dyer,16th Baronet KBE FRS ,1927.8.2,英国数学家 6

Figure 6: BSDß➂ Birch and Swinnerton-Dyer conjecture ✗L(C, s) := Q p|2δ (1 − app −s + p 1−2s ) −1 . Birch and Swinnerton-Dyer Conjecture. The Taylor expansion of L(C, s) at s = 1 has the form L(C, s) = c(s − 1)r + higher order terms with c 6= 0 and r = rank(C(Q)). Bryan John Birch F.R.S.➜1931.9.25➜❂■ê➷❬. Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS ➜1927.8.2➜❂■ê➷❬. 6